In contrast to the formation energy of vacancies, there exists no direct measurement for the formation energy of self-interstitials. We have mentioned in Section 1.01.2 that the displacement energy required to create a Frenkel pair is much larger than the combined formation energies of the vacancy and the self-interstitial. As pointed out, there exist a large energy barrier to create the Frenkel pair, namely the displacement energy Td, and this barrier is mainly associated with the insertion of the intersti­tial into the crystal lattice. However, although this barrier should be part of the energy to form a self­interstitial, it is by convention not included. Rather, the formation energy of a self-interstitial is consid­ered to be the increase of the internal energy of a crystal with this defect in comparison to the energy of the perfect crystal. In contrast, since vacancies can be created by thermal fluctuations at surfaces, grain boundaries, and dislocation cores by accepting an atom from an adjacent lattice site and leaving it vacant, no similar barrier exists. The activation energy for this process is simply the sum of the actual formation energy EV and the migration energy Em, that is, the energy for self-diffusion, QsD.

When Frenkel pairs are created by irradiation at cryogenic temperatures, self-interstitials and va­cancies can be retained in the irradiated sample. Subsequent annealing of the sample and measuring the heat released as the defects migrate and then disappear provide an indirect method to measure the energies of Frenkel pairs. Subtracting from these calorimetric values, the vacancy formation energy should give the formation energy of self-interstitials. The values so obtained for Cu7 vary from 2.8 to

4.2 eV, demonstrating just how inaccurate calorimet­ric measurements are. Besides, measurements have only been attempted on two other metals, Al and Pt, with similar doubtful results. As a result, theoreti­cal calculations or atomistic simulations provide per­haps more trustworthy results.

For a theoretical evaluation of the formation energy, we can consider the self-interstitial as an inclusion (INC) as described in Appendix A. Accord­ingly, a volume O of one atom is enlarged by the amount A V or in other words, is subject to the trans­formation strain

ej = rb{j where 3r = AV/Q [30]

The energy associated with the formation of this inclusion is given in Table A2, and it can be written as

rr _ 2KmQ fAv2

U = 3K + 4m Q

This expression for the so-called dilatational strain energy provides a rough approximation to the forma­tion energy of a self-interstitial in fcc metals when the above volume expansion results are used.

However, as the nearest neighbor cells depicted in Figures 13 and 14 show, their distorted shapes can­not be adequately described with a radial expansion of the original cell in the ideal crystal as implied by eqn [31]. As the detailed analysis by Wolfer19 indicates, the [001] dumbbell interstitial in the fcc lattice does not change the cell dimension in the [001] direction. In fact, it shortens it slightly, imply­ing that e33 = —0.01005. The volume change is there­fore due to the nearest neighbor atoms moving on average away from the dumbbell axis. This can be represented by the transformation strain components en = e22 being determined by

AV

2en + Є33 = — = 1.10164

The transformation strain tensor for the [001] self­interstitials in fcc crystals is then

and it can be divided into an dilatational part, and a shear part, ~y, as shown.

To find the transformation strain tensor for the [011] self-interstitial in bcc crystal, it is convenient to use a new coordinate system with the x3-axis as the dumbbell axis and the x1- and x2-axes emanating from the midpoint of the dumbbell axis and pointing toward the corner atoms. While the distance between these corner atoms and the central atom in the original bcc unit cell is the interatomic distance r0, in the distorted cell containing the self-interstitial, their distance is reduced to v/3r0/2 as shown by Wolfer.19 As a result,

P3

єн = Є22 = — y — 1 ~ -0.134

the remaining strain component is then determined by DV

2eu + Є33 = о = 0.6418

The strain energy associated with the shear part can be shown (see Mura20) to be

rr 2(9K+8m)mOr

U1 12

1 5(3K + 4m) 2

where /2 = —e11 ?22 — ?22?33 — ?33?П [34]

and is equal to /2fcc = 0.1068 and /2bcc = 0.3633 for self­interstitials in fcc and bcc metals, respectively.

We consider now the total strain energy as a reasonable approximation for the formation energy of self-interstitials, namely

Ef « U = U0 + U1 [35]

The dilatational and the shear strain energies for some elements are listed in Table 7 in the fourth and fifth column, respectively. For the fcc elements, the ratio of Ux/U0 is about 0.2. In contrast, for the bcc elements (in italics) this ratio is about two.

To formulate quantitative or often even qualitative models for defect processes in materials, it is essential that lattice relaxation be effected. Without lattice relaxation, the total energies calculated for defect reactions would be so great that we would have to conclude that no point defects would ever form in the material.9

Each defect has an associated defect volume. That is, each defect, when introduced into the lattice, causes a distortion in its surroundings, which is
manifested as a volume change arising as a result of the way in which the lattice responds, that is, how the lattice ions relax around the defect. For example, vacancies in ionic materials usually result in positive defect volumes. Consider the example of a vacancy in MgO. The nearest neighbor cations are displaced outward, away from the vacant site, causing an increase in volume, whereas the second neighbor anions move inward, albeit to a lesser extent (see Figure 8).

What drives these ion relaxations is the change in the Coulombic interactions due to defect formation. We say that the oxygen vacancy carries an effective positive charge because an O2~ has been removed and thus, an electrostatic attraction between O2~ and Mg2+ is removed. As ionic forces are balanced in a crystal, the outer O2~ ions now attract the Mg2+ away from the VO* defect site. In covalent materials, vacant sites result in atomic relaxations that are due to the formation of an incomplete complement of bonds, often termed ‘dangling bonds.’ In this case, the net result can be different from those in ionic solids and by way of an example, in silicon, a vacancy results in a volume decrease. On the other hand, an arsenic substitutional atom causes an increase in vol — ume.10 Finally, in a material such as ZrN or TiN, which exhibits both covalent and metallic bonding, the volume of a nitrogen vacancy is practically zero.1

Clearly, the overall response of the lattice can be rather complicated. However, the defect volume can be determined fairly easily by applying the rela­tionship

Figure 8 Schematic of the lattice relaxations around an oxygen vacancy in MgO.

where uP is the defect volume (in A3); KT the iso­thermal compressibility (in eV/A3); V the volume of the unit cell (in A3); andfy the Helmholz free energy of formation of the defect (in eV).

Finally, defect associations can also (but not nec­essarily) have a significant effect on defect volumes for a given solution reaction. For example, for the Al2O3 solution in MgO if we assume isolated AlMg and VMMg has the least effect on lattice parameter as a function of AlMg whereas the formation of neutral AlMg • VMMgX has the greatest effect (in fact ten times the reduction in lattice parameter).12

Network dislocation structures are routinely observed in metals5,8,200 and ceramics300,301 irradiated at
temperatures above recovery Stage I to temperatures in excess of recovery Stage V. During prolonged irradiation, the microstructural evolution typically involves formation and growth of faulted dislocation loops, loop unfaulting to create perfect dislocation loops, and then loop interaction/impingement to form network dislocation structures. The network dislocations are typically randomly distributed and are often heavily jogged as opposed to the relatively straight dislocations found in unirradiated metals. Figure 33 shows a typical network dislocation micro­structure for irradiated copper.30 The quantitative value of the dislocation density can vary significantly among different materials within the same crystal structure. For example, typical network dislocation densities in irradiated metals at temperatures between recovery Stages III and V range from -0.01 to-0.1 x 1014m~2 for Cu302-304 to-1-10 x 1014 m 2

for pure Ni304 and austenitic stainless steel.20

SiC is an important engineering ceramic because of its high-temperature stability, high thermal conduc­tivity, and special electronic properties. It has been proposed for use in nuclear applications including structural components in fusion reactors, cladding material for gas-cooled fission reactors, and as an inert matrix for the transmutation of plutonium and other transuranics.32 In high-temperature gas-cooled reactors, SiC is the primary barrier material for TRISO coated fuel particles.33 Also, SiC fiber, SiC matrix (SiC/SiC) composites are attractive candidate materials for first wall and blanket components in fusion reactors.34

Only limited studies of elevated-temperature microstructural evolution (dislocation loops, voids, etc.), based on neutron or ion irradiations, have been performed on SiC. In pyrolytic p-SiC (cubic, 3C), Price35 found small (2-5 nm diameter) {111} Frank loops following neutron irradiation at 900 °C to 2.4 x 1021 n cm~2 (E > 0.18 MeV) (-2.4 dpa). Yano and Iseki36 found the same loops in p-SiC irra­diated at 640°C to 1.0 x 1023 ncm~2 (E> 0.10MeV) (-100 dpa) and, using high-resolution TEM, deter­mined these to be 1/3 (111) {111} interstitial Frank loops. These loops are constructed by inserting a single extra Si-C layer into the CABCAB Si-C stacking sequence. This produces the sequence CA|C’B’|CAB, where the prime denotes a p rotation of the tetrahe­dral unit (note that an adjacent Si-C layer is modified by the insertion of the extra Si-C layer).

In 6H-type hexagonal a-SiC, Yano and Iseki36 found ‘black spot’ defects lying on (0001) planes following neutron irradiation at 840 ° C to 1.7 x 1021 ncm~2 (E > 0.10 MeV) (-1.7 dpa). They coarsened these defects using high-temperature annealing and determined the defects to be interstitial Frank loops. The stacking sequence along (0001) in 6H a-SiC is ABCA’B’C’. Yano and Iseki proposed that the Frank loops are formed by a mechanism similar to p-SiC (described above), wherein insertion of an extra Si-C layer modifies an adjacent Si-C layer to produce a sequence such as ABC|B’A’|C’B’. Such a defect is described as a 1/6 [0001] (0001) interstitial Frank loop.

For low temperatures (150-800 °C), small amounts of swelling (0-2%) are observed in monolithic SiC samples produced by chemical vapor deposition (CVD).33 It should be noted that CVD-SiC is cubic and highly faulted.37 This swelling saturates at low damage levels (a few dpa) and the saturated swelling is lower, the higher the temperature. Much of this swelling is due to strain caused by surviving interstitials formed during ballistic damage cascades. As the irradiation temperature approaches 1000 °C, the surviving defect fraction diminishes because interstitial mobility increases with temperature and i-v recombination is enhanced. Newsome eta/.33 found swelling values of 1.9, 1.1, and

0. 7% for neutron irradiations at 300, 500, and 800 °C, respectively.

Above 1000 °C, neutron irradiation-induced void formation in p-SiC was first observed by Price35 at 1250 °C (4.3-7.4 dpa) and 1500 °C (5.2-8.8 dpa). Interestingly, no dislocation loops were observable by TEM in these samples. Price35 postulated that the interstitials may have been annihilated at stacking faults. Alternatively, he suggested that interstitial defects were present following irradiation, but they were too small and the contrast too weak to detect them. Nevertheless, at present it is not clear whether void formation in SiC is due to vacancy supersatura­tion produced by a dislocation bias. In any case, swelling of a few percent was observed for irradia­tions at temperatures greater than 1000 °C, and Price38 speculated that this swelling probably does not saturate with dose.

At low temperatures (-60 °C), Snead and Hay37 observed both a — and p-SiC to amorphize following a total fast neutron fluence of 2.6 x 1021ncm~2 (-2.6 dpa). This amorphization transformation was accompanied by a large reduction in density

10.8%), that is, volumetric swelling of nearly 11%. Snead and Hay37 estimated that the critical tempera­ture for amorphization (the temperature above which amorphization is not possible) is 125 °C (a lower limit for the threshold amorphization temperature). The critical temperature is dose rate dependent. In the study above, the dose rate was —8 x 10-7dpas~3. In other electron and ion irradiation experiments with dose rates of — 1 x 10-3dpas_1, researchers found critical temperatures ranging from 20 to 70 °C for 2 MeV electron irradiations,39-41 150 °C for ener­

getic Si ions,42 and -220 °C for 1.5 MeV Xe ions.43,44

Edge dislocations contain jogs as illustrated by Figure 21. Here, the edge dislocation is the termina­tion of an atomic plane, and the last row of atoms, rendered in a darker color, delineates the dislocation line. The step in this line forms a jog. The two atomic planes next to the terminated atomic plane are shown in the upper part with perfect occupancies of all atomic positions, except that they bend along the dislocation line to become neighboring planes above the dislocation.

A vacancy may now form by an atom adjacent to the jog jumping to the terminating plane and thereby moving the jog by one atomic distance, as illustrated in the lower part of Figure 21. The vacancy created

may then diffuse away and into the surrounding crystal. Conversely, a vacancy that migrates to this jog will take the place of the atom next to the jog. As a result, the jog is simply displaced in the opposite direction. The absorption or emission of a vacancy restores the configuration of the edge dislocation.

The absorption or emission of a vacancy from a perfect core site of an edge dislocation is also possi­ble. However, the formation of a double jog would involve a large energy change, and for this reason, vacancy emission and absorption is unlikely for any site along the edge dislocation other than at already existing jogs.

Assuming local thermodynamic equilibrium for the region around a jog, the thermal vacancy concen­tration can be found as follows. We imagine that we create a vacancy in the surrounding lattice and add the extra atom to the jog. The total change in Gibbs free energy is then given by

A G = -(f — TsV ) + kT’n[W(NV + 1)] — kT’n[W(NV)] [91] since we added the Gibbs free energy (EV — Tf) to the crystal to create the vacancy but gained some configurational entropy of the amount k (N + NV + 1)!NV! "(N V + 1)!(N+N V)!

[92]

Here, N is the number oflattice sites per unit volume and Nv the number of vacancies, the latter being much smaller than N.

The creation of an additional vacancy in the vicin­ity of the dislocation jog occurs as a result of thermal fluctuation, and the Gibbs free energy change is on average equal to zero for this process. The condition AG = 0 defines the vacancy concentration in local thermodynamic equilibrium with the edge disloca­tion, and one obtains f IkT

Two minor but nevertheless important energy con­tributions have been neglected in the above deriva­tion. First, we did not consider the interaction of the vacancy with the stress field of the dislocation. Sec­ond, we also neglected any volume change of the crystal with the formation or absorption of a vacancy. A volume change will give rise to a work term if an external stress field aj is present in addition to the dislocation stress field.

The first contribution can be taken into account by adding to the vacancy formation energy the

interaction energy EI(ro, ‘), which was discussed in Section 1.01.5 and is further elaborated in Section 1.01.8. This energy is evaluated at the core radius rCj, and it varies with the polar angle ‘.

To evaluate the second contribution, we note that if vacancies were persistently absorbed at an edge dislocation, it would eventually eliminate the incom­plete crystal plane of atoms. The crystal volume would therefore contract in the direction perpendic­ular to this incomplete plane by the amount of the Burgers vector component b, in this direction. If tj = ajbj|b is the force per unit area on the incom­plete plane as exerted by the external stress field aj, then the work done against the external loads is

-8Ab, t, =-8A b, a°t-bj|b [94]

when the area 8A is eliminated from the incomplete plane. The negative sign turns into a positive one if vacancies are emitted from the dislocation and if it climbs to expand the area of the incomplete plane. For one vacancy emitted and hence one atom added to the incomplete plane, 8Ab = O, and the energy gained from the work expended by the external loads is

,2) a°Mbj + = Os? + [95]

The second term is due to the fact that the crystal does not expand by the entire atomic volume but is reduced by the vacancy relaxation volume. Note that VR has in general a negative value, and that it results in an isotropic volume change; hence, the hydrostatic stress performs the work.

As a result of these two additional contributions one finds that the local equilibrium vacancy concen­tration at a dislocation is

EI(rC.’) VvaH. Oa? kT + kT + kT

We see that the interaction energy £I(rC,’) gives a spatial dependence to the local equilibrium vacancy concentration. In addition, external forces on the crys­tal or stresses produced at the location of the disloca­tion by other distant defects will further change the local equilibrium concentration.

Displacement damage can occur in materials when the energy transferred to lattice atoms exceeds a critical value known as the threshold displacement energy (Ed), which has a typical value of 30-50 eV.8,18,40 Figure 2 shows an example of the effect of bombard­ing energy on the microstructure of CeO2 during electron irradiation near room temperature.41 The loop density increases rapidly with increasing energy
above 200 keV, suggesting that 200 keVelectrons trans­fer elastic energy that is slightly above the threshold displacement energy. High-resolution microstructural analysis determined that the dislocation loops were associated with aggregation of oxygen ions only (i. e., no Ce displacement damage) for electron energies up to 1250 keV, whereas perfect interstitial-type disloca­tion loops were formed for electron energies of 1500 keV and higher. Therefore, the corresponding displacement energies in CeO2 are ^30 and ^50 eV for the O and Ce sublattices, respectively.41

A wide range of PKA energies can be achieved during irradiation, depending on the type and energy of irradiating particle. For example, the average PKA energies transferred to a Cu lattice for 1 MeV electrons, protons, Ne ions, Xe ions, and neutrons are 25 eV, 0.5 keV, 9 keV, 50 keV, and 45 keV, respec — tively.42 Irradiation of materials with electrons and light ions introduces predominantly isolated SIAs and vacancies (together known as Frenkel pairs) and small clusters of these point defects, because of the low average recoil atom energies of ~0.1-1 keV. Con­versely, energetic neutron or heavy ion irradiations produce energetic displacement cascades that can lead to direct formation of defect clusters within isolated displacement cascades due to more ener­getic average recoil atom energies that exceed 10 keV. Figure 3 compares the weighted PKA energy values for several irradiation species.40,42

These differences in PKA energy produce signifi­cant changes in primary damage state that can have a pronounced effect on the microstructural evolution observed during irradiation. As briefly mentioned in Section 1.03.3.1, the defect production efficiency per dpa determined from electrical resistivity mea­surements during irradiation near absolute zero and MD simulation studies is significantly lower (by about

a factor of 3-4) for energetic displacement cascade conditions compared to isolated Frenkel pair condi­tions, due to pronounced in-cascade recombination and clustering processes.38,39 MD computer simula — tions43—46 and in situ or postirradiation thin foil exper­imental studies13,14,47,48 (where interaction between different displacement damage events is minimal due to the strong influence of the surface as a point defect sink) have found that defect clusters visible by transmission electron microscopy (TEM) can be produced directly in displacement cascades if the average PKA energy exceeds 5-10 keV. Irradiations with particles having significantly lower PKA ener­gies typically produce isolated Frenkel pairs and sub­microscopic defect clusters that can nucleate and coarsen via diffusional processes. The microstructural evolution of an irradiated material is controlled by different kinetic equations if initial defect clustering occurs directly within the displacement cascade (~0.1—1 ps timescale) versus three-dimensional ran­dom walk diffusion to produce defect cluster nucle — ation and growth, particularly ifsome ofthe in-cascade created defect clusters exhibit one-dimensional glide 49-52 As discussed in Chapter 1.13, Radiation Damage Theory, this can produce significant differ­ences in the microstructural evolution for features such as voids and dislocation loops. Figure 4 compares the microstructure produced in copper following irra­diation near 200 °C with fission neutrons53 and 1 MeV electrons.54,55 Vacancies and SIAs are fully mobile in copper at this temperature. The 1 MeV electron pro­duces a steady flux of point defects that leads to the

creation of a moderate density of large faulted interstitial loops. On the other hand, the creation of SFTs and small dislocation loops directly in fission neutron displacement cascades creates a high density (~2 x 1023 m~ ) of small defect clus­ters, and the high point defect sink strength asso­ciated with these defect clusters inhibits the growth of dislocation loops. As shown in Figure 4, the net result is a dramatic qualitative and quantitative

difference in the irradiated microstructure due to differences in the PKA spectrum.

Electron microscopy48’56 and binary collision48’57 and MD simulation45 studies have found that irradi­ation with PKA energies above a critical material — dependent value of ^ 10-50 keV results in formation of multiple subcascades (rather than an ever — increasing single cascade size), with the size of the largest subcascades being qualitatively similar to an isolated cascade at a PKA energy near the critical value. Figure 5 compares MD simulations of the peak displacement configurations of PKAs in iron with energies ranging from 1 to 50 keV.58 At low PKA energies’ the size of the displacement cascade increases monotonically with PKA energy. When the PKA energy in Fe exceeds a critical value of ~10 keV, multiple subcascades begin to appear, with the largest subcascade having a size comparable to the 10 keV cascades. The number of subcascades increases with increasing PKA energy, reaching ^5 subcascades for a PKA energy of 50 keV in Fe. A fortunate conse­quence of subcascade formation is that fission reactor irradiations (~1 MeV neutrons) can be used for ini­tial radiation damage screening studies of potential future fusion reactor (~14 MeV neutrons) materials, since both would have comparable primary damage subcascade structures.59,60 Further details on the ef­fect of PKA spectrum on primary damage formation
are given in Chapter 1.11, Primary Radiation Damage Formation.

An interesting phenomenon is observed when irra­diated alloys are tested at temperatures different from the irradiation temperature. Figure 12 shows total elongation data from cold-worked type 316 stainless steel irradiated to displacement levels of 48-63 dpa in the FFTF, where elongation is plotted
against the increment of the test temperature above the irradiation temperature.2 Although there is significant scatter in the data, the elongations below 1% obtained by test temperatures about 100 °C above the irradiation temperature are cause for concern. This phenomenon has also been observed in higher nickel alloys. The cause of this phenome­non remains elusive, pending further testing with

different holding times at various temperatures before tensile testing. Migration of interstitial solutes to moving dislocations is a candidate mechanism for this phenomenon.

The primary motivation for the earliest research was the observation that even a small concentration of bulk He, in some cases in the range of one appm or less, generated in fission reactor irradiations of AuSS, could lead to HTHE, manifested as significant re­ductions in tensile and creep ductility and creep rupture times. The degradation of these properties coincided with an increasing transition from trans­granular to intergranular rupture.10,85-89 HTHE is attributed to stress-driven nucleation, growth, and coalescence of grain boundary cavities formed on the He bubbles. The early studies included mixed spec­trum neutron irradiations that produce large amounts of He in alloys containing Ni and B. Figure 6 shows one extreme example of the dramatic effect of HTHE on creep rupture times for a 20% cold-worked (CW) 316 stainless steel tested at 550°C and 310MPa following irradiations between 535 and 605 °C in the mixed spectrum HFIR that produced up to 3190 appm He and 85 dpa.88 At the highest He concentration, the creep rupture time is reduced by over four orders of magnitude, from several thousand to less than 0.1 h. A comprehensive review of the large early body of research on He effects on mechanical properties of AuSS can be found in Mansur and Grossbeck.11

The early fission reactor irradiations research on HTHE was later complemented by extensive accelerator-based He ion implantation experiments, primarily carried out in the 1980s (see Schroeder and Batfalsky90 and Schroeder, Kesternich and Ullmaier91 as examples) but that have continued to recent times.92 HTHE models were developed during this period, primarily in conjunction with the He ion implanta­tion experiments.93-100 The He implantation studies and models are discussed further in Sections 1.06.3.6 and 1.06.3.7. A more general review of He effects, again primarily in AuSS, can be found in Ullmaier99 and a comprehensive model-based description of the behavior of He in metals in Trinkaus.96

Research on He effects was also greatly stimulated by the discovery of large growing voids in irradiated AuSS.1 As an example, Figure 7(a) shows swelling curves for a variety of alloys used in reactor applica — tions.102-104 Figure 7(b) illustrates macroscopic conse­quences of this phenomenon in an AuSS.105 Figure 8 shows a classical micrograph of a solution annealed (SA) AuSS with dislocation loops and line segments, preci­pitates, precipitate-associated and matrix voids, and possibly He bubbles (the small cavities). RT-based modeling studies of void swelling began in the early 1970s,106,107 peaking in the 1980s, and continuing up to recent times.108 Most of the earliest models emphasized the complex effects of He on void swelling. ,

As discussed in more detail below, these and later models rationalized many observed swelling trends and also suggested approaches to developing more swelling-resistant AuSS, largely based on trapping

He in small bubbles at the interfaces of fine-scale precipitates. Reviews summarizing mechanisms and modeling of swelling carried out during this period, including the role of He, can be found in Odette,1 Odette, Maziasz and Spitznagel,1 Mansur,11 Mansur and Coghlan,114 Freeman,115 and Mansur.116

Reviews of experimental studies of void swelling can be found in later studies by Maziasz16 and Zinkle, Maziasz and Stoller.117

Further motivation for understanding He effects was stimulated by a growing interest in the effects of the very high transmutation levels produced in fusion reactor spectra (see Section 1.06.1).89,99,111,112,118 Experimental studies comparing microstructural evolutions in AuSS irradiated in fast (lower He) and mixed spectrum (high He) reactors provided key insight into the effects of He.16,119,120 Helium effects were also systematically studied using dual-beam He-heavy ion CPI.26,121-129

Beginning in the mid-1970s, a series of studies specifically addressed the critical question of how to use fission reactor data to predict irradiation effects in fusion reactors,15,109-112,118,130-133 and this topic remains one of intense interest to this day. An indica­tion of the complexity of He effects is illustrated in Figure 9, showing microstructures in a dual-beam He-heavy ion irradiation of a SA AuSS to 70 dpa and 625 °C at different He/dpa.1 3 In this case, voids do not form in the single heavy ion irradiation without He. At intermediate levels, of 0.2 appm/dpa, large voids are observed, resulting in a net swelling of 3.5%. At even higher levels of 20appm/dpa, the voids are more numerous, but smaller, resulting in less net swelling of 1.8%. These observations show that some He promotes the formation of voids, but that higher amounts can reduce swelling. Figure 10 shows the effect of various conditions for

introducing 1400 appm He coupled with a 4MeV Ni ion irradiation of a swelling-prone model SA AuSS to 70 dpa at 625 °C.125 In this case, the swelling is largest («18% due to voids) with no implanted He and smallest («1%) with He preimplanted at ambient temperature due to the very high density of bub­bles. These results also show that voids can form

at sufficiently high CPI damage rates without He, probably assisted by the presence of impurities like oxygen and hydrogen. Most notably, however, the swelling decreases with increasing bubble num­ber densities.

The emphasis of more recent experimental work has been on SPNI that generate large amounts

of He, compared with fission reactors, as well as dis­placement damage (see Section 1.06.4). The SPNI studies have focused on mechanical properties and microstructures, primarily at lower irradiation tem­peratures, nominally below the HTHE regime. In addition, as discussed in Section 1.06.2, a previously proposed in situ He injection technique31,49 has recently been developed and implemented to study He-displacement damage interactions in mixed spec­trum reactor irradiations (e. g., HFIR) at reactor­relevant dpa rates.23,51-53 As discussed in Section 1.06.5, recent modeling studies have emphasized electronic and atomistic evaluations of the energy parameters that describe the behavior of He in solids, including interactions with point and extended defects134-136 (and see Section 1.06.5). The refined parameters are being used in improved RT and Monte Carlo models of He diffusion and clustering to form bubbles on dislocations, precipitates, and GBs, as well as in the matrix, as discussed in Section 1.06.6.

It is again important to emphasize that the broad framework for predicting He effects is an under­standing and modeling of its generation, transport, and fate, as well as the multifaceted consequences of this fate. We begin with a discussion of the role of He in void swelling and other microstructural evo­lution processes. We then return to the issue of HTHE.

A1 An Effective Medium Approximation

The crystal defects that we consider here are mainly of microscopic size ranging from single atoms to

clusters of several hundreds of atomic defects, and the volume fraction they occupy ranges from a part per million to perhaps a tenth of a percent. So if p is the typical radius of a defect under consideration and 2R the average distance between them, then the defect volume fraction

S = (p/R)3 [A1]

is a small number. As a result, each defect is sur­rounded by a cell that contains no other defect, and the solid can be viewed as composed of these cells. If the solid as a whole is free of external stresses on its macroscopic surface, then the normal stress component averaged over the surface of each cell is also zero, since on the cell boundary, the stress fields of two neighboring defects overlap and cancel on average. In the final step in this cellular approach, we approximate the typical cell by a spherical region with radius R with a defect of radius p at its center. It is this effective medium approximation that elevates the analysis of a sphere with a defect at its center to more than just an academic exercise, and its results are more generally valid for defects in solids of finite extent.

Atomistic computer simulations using either semi-empirical potentials or first-principle methods are usually carried out in a periodically repeated cell that contains a finite number of atoms. However, in this case the medium is in fact infinite, and the results are different from those obtained for a finite crystal on whose external surface the stress field created by the defect satisfies the boundary condition of zero tractions, that is, the vanishing of the stress compo­nent normal to the external surface. Satisfying this boundary condition is different from having a defect stress field that approaches zero at infinity. This difference is often attributed to the so-called image stresses that exist in a finite crystal, but not in an infinite one. The model of a defect at the center of a finite sphere ofradius R captures this difference, and as we shall see, this difference becomes independent of the radius R as it approaches infinity but such that S, the defect concentration, remains small but constant.

Due to relatively large concentrations of conduction electrons, materials with metallic bonding typically do not exhibit sensitivity to ionizing radiation. On the other hand, semiconductor and insulating materials can be strongly affected by ionizing radiation by vari­ous mechanisms that lead to either enhanced or sup­pressed defect accumulation.159 Some materials such as alkali halides, quartz, and organic materials, are susceptible to displacement damage from radiolysis reactions.65,160-163 In materials that are not suscepti­ble to radiolysis, significant effects from ionizing radi­ation can still occur via modifications in point defect migration behavior. Substantial reductions in point defect migration energies due to ionization effects have been predicted, and significant microstructural changes attributed to ionization effects have been observed in several semiconductors and inorganic insulator materials.18,159,164-169 The effect of ionizing radiation can be particularly strong for electron or light ion beam irradiations of certain ceramic materi­als since the amount of ionization per unit displace­ment damage is high for these irradiation species; the ionization effect per dpa is typically less pronounced for heavy ion, neutron, or dual ion beam irradiation. Figure 20 summarizes the effect of variations in the ratio of ionizing to displacive radiation (achieved by varying the ion beam mass) on the dislocation loops

density and size in several oxide ceramics.94,169,170

The loop density decreases rapidly when the ratio of ionizing to displacive radiation (depicted in Figure 20 as electron-hole pairs per dpa) exceeds a material — dependent critical value, and the corresponding loop size simultaneously increases rapidly.

Numerous microstructural changes emerge in mate­rials irradiated with so-called swift heavy ions that produce localized intense energy deposition in their ion tracks. Defect production along the ion tracks is observed above a material-dependent threshold value for the electronic stopping power with typical values of 1-50keVnm-1.159,171-175 The microstructural changes are manifested in several ways, including dislocation loop punching,176 creation of amorphous tracks with typical diameters of a few nm,159,173,174,177-180

atomic disordering,176,181,182 crystalline phase transfor-

mations,171 destruction of preexisting small dislocation

50

40

30

20

10

Figure 20 Effect of variations in ionizing to displacive radiation on the dislocation loop density and size in ion-irradiated MgO, Al2O3, and MgAl2O4. Adapted from Zinkle, S. J. Radiat Eff. Defects Solids 1999, 148, 447-477; Zinkle, S. J. J. Nucl. Mater. 1995, 219, 113-127; Zinkle, S. J. In Microstructure Evolution During Irradiation; Robertson, I. M., Was, G. S., Hobbs, L. W., Diaz de la Rubia, T., Eds. Materials Research Society: Pittsburgh, PA, 1997; Vol. 439, pp 667-678.

_p

(a) 20 nm

Figure 21 Plan view microstructure of disordered ion tracks in MgAl2O4 irradiated 430 MeV Kr ions at room temperature to a fluence of 6 x 1015 ions per square meter (isolated ion track regime) under (a) weak dynamical bright field and (b) g = (222) centered dark field imaging conditions (tilted 10° to facilitate viewing of the longitudinal aspects of the ion tracks). High-resolution TEM and diffraction analyses indicate disordering of octahedral cations (but no amorphization) within the individual ion tracks. Adapted from Zinkle, S. J.; Skuratov, V. A. Nucl. Instrum. Methods B 1998, 141(1-4), 737-746;

Zinkle, S. J.; Matzke, H.; Skuratov, V. A. In Microstructural Processes During Irradiation; Zinkle, S. J., Lucas, G. E.,

Ewing, R. C., Williams, J. S., Eds. Materials Research Society: Warrendale, PA, 1999; Vol. 540, pp 299-304.

loops,176 and formation of nanoscale hillocks and sur­rounding valleys183’184 at free surfaces. Annealing of point defects occurs for irradiation conditions below the material-dependent threshold electronic stopping power for track creation,159,180,185,186 whereas defect production occurs above the stopping power threshold.159,171,173,175,178,180,183,185,186 The swift heavy ion annealing and defect production phenomena are

observed in both metals and alloys171,175,183,185,186 as well as nonmetals.159,172,173,178-180,187-190 Defect pro­duction by swift heavy ions is of importance for
understanding the radiation resistance of current and potential fission reactor fuel systems, including the mechanisms responsible for the finely polygo — nized rim effect18 ,191 in UO2 and radiation stability of inert matrix fuel forms.182,189,191 The swift heavy ion defect production mechanism is generally attributed to thermal spike178,192 and self-trapped exciton187 effects. Figure 21 shows examples of the plan view (i. e. along the direction of the ion beam) microstructure of dis­ordered ion tracks in MgAl2O4 irradiated with swift

176,182

heavy ions.